Related papers: A Surface Degeneration With Non-Collapsible Contra…
Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne0$, or into…
We study the weighted spectrum and vanishing cohomology for several classes of isolated hypersurface singularities, and how they contribute to the limiting mixed Hodge structure of a smoothing. Applications are given to several types of…
We introduce a new obstruction to the existence of a universal $0$-cycle on a smooth projective complex variety. As an application, we construct a smooth projective complex surface whose Chow group of $0$-cycles is representable but which…
In this paper we study the Bruhat decomposition of not necessarily connected reductive quasi-split groups $G$ with respect to not necessarily connected parabolic subgroups. If $G$ is defined over a finite field, we construct a smooth…
We study deformation theory of elliptic fibre bundles over curves in positive characteristics. As applications, we give examples of non-liftable elliptic surfaces in charactertic two and three, which answers a question of Katsura and Ueno.…
We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard…
We implement a differential-geometric approach to normal forms for contracting measurable cocycles to $\mbox{Diff}^q({\bf R}^n, {\bf 0})$, $q \geq 2$. We obtain resonance polynomial normal forms for the contracting cocycle and its…
We show the existence of several new families of non-compact constant mean curvature surfaces: (i) singly-punctured surfaces of arbitrary genus $g \geq 1$, (ii) doubly-punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.
We prove that for a Q-Gorenstein degeneration $X$ of del Pezzo surfaces, the number of non-Du Val singularities is at most $\rho(X)+2$. Degenerations with $\rho(X)+2$ and $\rho(X)+1$ non-Du Val points are investigated.
This paper investigates the geometry and singularities of parallel surfaces of cuspidal cross caps, the fundamental non-front frontal singularities. We establish a criterion for the degeneracy of the distance squared function in terms of…
For a genus g handlebody H a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H, is constructed and several properties are established. In particular, this complex naturally contains, as a…
We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena…
The analysis of 3D symmetric second-order tensor fields often relies on topological features such as degenerate tensor lines, neutral surfaces, and their generalization to mode surfaces, which reveal important structural insights into the…
The scaling properties of one-dimensional deconstructed surfaces are studied by numerical simulations of a disaggregation model. The model presented here for the disaggregation process takes into account the possibility of having quenched…
Minimal surfaces are ubiquitous in nature. Here they are considered as geometric objects that bear a deformation content. By refining the resolution of the surface deformation gradient afforded by the polar decomposition theorem, we…
For a subvariety of a smooth projective variety, consider the family of smooth hypersurfaces of sufficiently large degree containing it, and take the quotient of the middle cohomology of the hypersurfaces by the cohomology of the ambient…
We extend the doubly degenerate Cahn-Hilliard (DDCH) models for isotropic surface diffusion, which yield more accurate approximations than classical degenerate Cahn-Hilliard (DCH) models, to the anisotropic case. We consider both weak and…
We introduce tropical complexes, as an enrichment of the dual complex of a degeneration with additional data from non-transverse intersection numbers. We define cycles, divisors, and linear equivalence on tropical complexes, analogous both…
Motivated by the geometry of Levi degenerate CR hypersurfaces, we define a pre-K\"ahler structure on a complex manifold as a pre-symplectic structure compatible with the almost complex structure, i.e. a closed (1,1)-form. Extending Freeman…
The 2-girth of a 2-dimensional simplicial complex $X$ is the minimum size of a non-zero 2-cycle in $H_2(X, \mathbb{Z}/2)$. We consider the maximum possible girth of a complex with $n$ vertices and $m$ 2-faces. If $m = n^{2 + \alpha}$ for…